Modules with Chain Conditions on S-Closed Submodules

Let L be a commutative ring with identity and let W be a unitary left L- module. A submodule D of an L- module W is called s- closed submodule denoted by D ≤sc W, if D has no proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D = H. In this paper, we study modules which satisfies the ascending chain conditions (ACC) and descending chain conditions (DCC) on this kind of submodules.

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W-Closed Submodule and Related Concepts

Ibn AL- Haitham Journal For Pure and Applied Science ◽

10.30526/31.2.1955 ◽

2018 ◽

Vol 31 (2) ◽

pp. 164

Author(s):

Haibat K. Mohammad Ali ◽

Mohammad E. Dahsh

Keyword(s):

Commutative Ring ◽

Chain Condition ◽

Essential Extension ◽

Basic Properties ◽

Closed Submodule ◽

Closed Submodules

Let R be a commutative ring with identity, and M be a left untial module. In this paper we introduce and study the concept w-closed submodules, that is stronger form of the concept of closed submodules, where asubmodule K of a module M is called w-closed in M, "if it has no proper weak essential extension in M", that is if there exists a submodule L of M with K is weak essential submodule of L then K=L. Some basic properties, examples of w-closed submodules are investigated, and some relationships between w-closed submodules and other related modules are studied. Furthermore, modules with chain condition on w-closed submodules are studied.

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N-th root rings

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013083 ◽

1987 ◽

Vol 35 (1) ◽

pp. 111-123 ◽

Cited By ~ 1

Author(s):

Henry Heatherly ◽

Altha Blanchet

Keyword(s):

Commutative Ring ◽

Subdirect Product ◽

General Construction ◽

Fixed Integer ◽

Chain Conditions

A ring for which there is a fixed integer n ≥ 2 such that every element in the ring has an n-th in the ring is called an n-th root ring. This paper gives numerous examples of diverse types of n-th root rings, some via general construction procedures. It is shown that every commutative ring can be embedded in a commutative n-th root ring with unity. The structure of n-th root rings with chain conditions is developed and finite n-th root rings are completely classified. Subdirect product representations are given for several classes of n-th root rings.

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Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

Glasnik Matematicki ◽

10.3336/gm.56.2.08 ◽

2021 ◽

Vol 56 (2) ◽

pp. 343-374

Author(s):

Boris Guljaš ◽

Keyword(s):

Hilbert Space ◽

Hilbert Modules ◽

Strict Topology ◽

Direct Sums ◽

Essential Ideal ◽

Closed Submodule ◽

Closed Submodules ◽

Hereditary Algebras ◽

Multiplier Algebras

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

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Some chain conditions on weak incidence algebras

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.2389 ◽

2005 ◽

Vol 2005 (15) ◽

pp. 2389-2397

Author(s):

Surjeet Singh ◽

Fawzi Al-Thukair

Keyword(s):

Commutative Ring ◽

Partially Ordered Set ◽

Nonempty Subset ◽

Ordered Set ◽

Partially Ordered ◽

Incidence Algebra ◽

Chain Conditions ◽

Incidence Algebras

LetXbe any partially ordered set,Rany commutative ring, andT=I∗(X,R)the weak incidence algebra ofXoverR. LetZbe a finite nonempty subset ofX,L(Z)={x∈X:x≤z for some z∈Z}, andM=Tez. Various chain conditions onMare investigated. The results so proved are used to construct some classes of right perfect rings that are not left perfect.

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When is every module with essential socle a direct sum of automorphism-invariant modules?

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501856 ◽

2019 ◽

Vol 19 (10) ◽

pp. 2050185

Author(s):

Shahabaddin Ebrahimi Atani ◽

Saboura Dolati Pish Hesari ◽

Mehdi Khoramdel

Keyword(s):

Commutative Ring ◽

Principal Ideal ◽

Essential Extension ◽

Principal Ideal Ring ◽

Von Neumann ◽

Simple Modules ◽

Direct Sum ◽

Finite Dimensional ◽

Ideal Ring ◽

Von Neumann Regular

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

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Modules whose closed submodules are finitely generated

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500005083 ◽

1991 ◽

Vol 34 (1) ◽

pp. 161-166 ◽

Cited By ~ 3

Author(s):

Nguyen V. Dung

Keyword(s):

Some Results on Pure Submodules Relative to Submodule

Baghdad Science Journal ◽

10.21123/bsj.12.4.833-837 ◽

2015 ◽

Vol 12 (4) ◽

pp. 833-837

Author(s):

Baghdad Science Journal

Keyword(s):

Commutative Ring ◽

Extending Module ◽

Closed Submodules ◽

Pure Submodules

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be pure relative to submodule T of M (Simply T-pure) if for each ideal A of R, N?AM=AN+T?(N?AM). In this paper, the properties of the following concepts were studied: Pure essential submodules relative to submodule T of M (Simply T-pure essential),Pure closed submodules relative to submodule T of M (Simply T-pure closed) and relative pure complement submodule relative to submodule T of M (Simply T-pure complement) and T-purely extending. We prove that; Let M be a T-purely extending module and let N be a T-pure submodule of M. If M has the T-PIP, then N is T-purely extending.

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On quasi-radical of near-ring of polynomials

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2019.56.2.1430 ◽

2019 ◽

Vol 56 (2) ◽

pp. 252-259

Author(s):

Ebrahim Hashemi ◽

Fatemeh Shokuhifar ◽

Abdollah Alhevaz

Keyword(s):

Commutative Ring ◽

Nilpotent Elements ◽

Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

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N-ideals of commutative rings

Filomat ◽

10.2298/fil1710933t ◽

2017 ◽

Vol 31 (10) ◽

pp. 2933-2941 ◽

Cited By ~ 3

Author(s):

Unsal Tekir ◽

Suat Koc ◽

Kursat Oral

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Proper Ideal ◽

Prime Ideals ◽

Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

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Derivations of differentially semiprime rings

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500797 ◽

2019 ◽

Vol 12 (05) ◽

pp. 1950079

Author(s):

Ahmad Al Khalaf ◽

Iman Taha ◽

Orest D. Artemovych ◽

Abdullah Aljouiiee

Keyword(s):

Commutative Ring ◽

Semiprime Ring ◽

Prime Rings ◽

Commutative Case ◽

Lie Ring ◽

Lie Rings ◽

Semiprime Rings ◽

Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

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